ps1 - S = xe-x Find a solution for the c ﬁeld as a...

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APPH4200 Physics of Fluids: Homework 1 Question 1 Consider the two-dimensional, incompressible ﬂow u = ( u,v,w ) = (cos αt, sin αt, 0) Give a complete description of the streamlines and the trajectory of a particle released at the origin at t = 0. Extra credit for the brave: Dye is injected into the ﬂow at the origin beginning at t = 0 and continuously thereafter. Describe the streaklines traced out by the dyed ﬂuid, for all time t 0. How does the streakline which initiates at the origin at t = 0 diﬀer from the trajectory which originates at the same place and time? Question 2 Consider the steady, two-dimensional, incompressible ﬂow u = ( u,v,w ) = ( - x,y, 0) on the half-plane (0 < x < , -∞ < y < ). 1. Sketch the streamlines for the ﬂow. 2. Compute the strain and rotation tensors (if it’s not easy, you’re doing it wrong). 3. A spatially distributed source releases a passive chemical tracer, with mixing ratio c , into the ﬂow. The tracer obeys dc dt = S, with S the source and d/dt = ∂/∂t + u · ∇ . In this case assume that

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Unformatted text preview: S = xe-x . Find a solution for the c ﬁeld as a function of x,y , assuming that it is in steady state. ( Hint : you will ﬁnd the equation much easier to solve if you ﬁrst make an educated assumption about the symmetry of the solution. You only need to ﬁnd a valid solution, not necessarily the most general.) 4. The steady state exists despite the fact that the tracer is being continuously added everywhere. How is the budget of tracer mass closed, that is, where is the dye going? 1 Question 3 Prove the identity u · ∇ u = ( ∇ × u ) × u + ∇ ( 1 2 | u | 2 ) (This is a vector equation with three components; it is suﬃcient to work out one component out of three). Question 4 Kundu & Cohen, Chapter 3, problem 6. Question 5 Kundu & Cohen, Chapter 4, problem 2. NOTE: I will re-use these questions. Please do not share your answers or the class solutions with students from previous or future classes! 2...
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ps1 - S = xe-x Find a solution for the c ﬁeld as a...

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